### Riemann-Roch Theory for Directed Graphs

- Series
- SIAM Student Seminar
- Time
- Friday, November 19, 2010 - 13:00 for 1 hour (actually 50 minutes)
- Location
- Skiles 255
- Speaker
- Spencer Backman – School of Mathematics, Georgia Tech

The talk will begin with an elementary geometric discussion
of Riemann-Roch theory for sub-lattices of the integer lattice
orthogonal to some positive vector. A pair of necessary and
sufficient conditions for such a lattice to have the Riemann-Roch
property will be presented. By studying a certain chip firing game on
a directed graph related to the lattice spanned by the rows of its
Laplacian I will describe a combinatorial method for checking whether
a directed graph has the Riemann-Roch property. The talk will
conclude with a presentation of arithmetical graphs, which after the
application of a simple transformation, may be viewed as a special
class of directed graphs. Examples from this class demonstrate that
either, both or neither of the Riemann-Roch conditions may be
satisfied for a directed graph. This is joint work with Arash Asadi.